The Minimal Model Program for the Hilbert Scheme of Points on P^2 and\n Bridgeland Stability

Abstract

In this paper, we study the birational geometry of the Hilbert scheme of n\npoints on P^2. We discuss the stable base locus decomposition of the effective\ncone and the corresponding birational models. We give modular interpretations\nto the models in terms of moduli spaces of Bridgeland semi-stable objects. We\nconstruct these moduli spaces as moduli spaces of quiver representations using\nG.I.T. and thus show that they are projective. There is a precise\ncorrespondence between wall-crossings in the Bridgeland stability manifold and\nwall-crossings between Mori cones. For n at most 9, we explicitly determine the\nwalls in both interpretations and describe the corresponding flips and\ndivisorial contractions.\n